Unique log file for each instance of class

I am currently running a windows service that creates multiple instances of a class.

At the top of the service class and every other class in my solution, I have something like this:

private static readonly ILog _log = LogManager.GetLogger(typeof(SomeClassTypeHere));

In my App.config, I have Log4Net configured for a single file:

<log4net debug="true">
    <appender name="RollingFileAppender" type="log4net.Appender.RollingFileAppender">
      <file value="Logs\SomeLogFileName.xml" />
      <appendToFile value="true" />
      <rollingStyle value="Size" />
      <countDirection value="1" />
      <maxSizeRollBackups value="30" />
      <maximumFileSize value="10MB" />
      <staticLogFileName value="true" />
      <lockingModel type="log4net.Appender.FileAppender+MinimalLock" />
      <layout type="log4net.Layout.XmlLayoutSchemaLog4j">
        <locationInfo value="true" />
      </layout>
    </appender>

    <root>
      <level value="INFO" />
      <appender-ref ref="RollingFileAppender" />
    </root>
  </log4net>

This works great in most respects, and everything logs to a single file. However, I'd really like to create a separate log file for each instance of a particular class that my service creates.
This is a class that we often need to monitor for support and we can have a handful of instances running at the same time.
We don't know which instances will be running at a given time, so it makes creating static files in the configuration kinda painful.

I tried taking off the readonly modifier and setting the following in the class constructor:

_log = LogManager.GetLogger("DataCollectionClass_" + deviceName + "_" + DateTime.Now.ToString("MMddyyyy"), typeof(SomeClassTypeHere));

But that req